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Task 1.2: Fully Coupled Hydrogeophysical Inversion of Salt-Tracer Experiments
RECORD PhD Retreat9th-10th June 2009
Davina Pollock, Center for Applied Geoscience, University of Tübingen / ETH Zürich
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Introduction Hydraulic conductivity (K) = key groundwater parameter Measurements:
Wells expensive low resolution Geophysics does not measure directly hydraulic conductivity
petrophysical relationships
Use geophysics to monitor hydraulic testsRelate change in geophysical properties to hydraulic stress Electrical Resistivity Tomography (ERT) during salt-tracer
tests Various previous studies
(e.g. Binley et al., 1996; Slater et al., 2000; Binley et al., 2002; Kemna et al., 2002; Vanderborght et al., 2005; Singha and Gorelick, 2005, Cassiani, 2006)
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
General Approach Standard: classical geophysical inversion followed by
hydraulic interpretation of concentration distribution decoupled risk of non-physical results (e.g., loss of mass) no direct inference of hydraulic conductivity
Approach in which hydraulics and geophysics are considered as a coupled system:
physics of groundwater flow, transport, and geoelectrics satisfied by definition
efficient and stable inversion strategy?
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Our Approach
Time-lapse ERT during tracer test Analyse temporal moments of electrical-potential
perturbations Relate mean arrival time of electrical signal to hydraulic
conductivity Temporal moment generating equations Incorporation into geostatistical inversion
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Temporal Moments of Concentration Definition of zeroth and first moments:
Advection Dispersion Equation:
Moment generating equation:
Already used in inversion(e.g. Harvey and Gorelick, 1995; Cirpka and Kitanidis 2000; James et al., 2000; Nowak and Cirpka, 2006)
0 cctc Dq
ck
ck
ck mkmm 1 Dq
01
00
,
,
dtttcm
dttcm
c
c
xx
xx
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Temporal Moments of Potential Perturbations Definition
Poisson equation
Initial distribution φ0
Linearisation perturbation φ‘
Moment generating equation
ioIc xxxx 0
00 c
00 ckk mm
ioI xxxx 00
0
, dtttm kk xx
(Pollock & Cirpka, 2008)
Valid if κc<<σ0 and for m1/m0
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Setup Quasi two-dimensional sandbox Injection of salt tracer ERT monitoring using different electrode configurations Results shown are for a hypothetical test case
Electrodes
Fixed head
Fixed head
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Measurements
tc=m1/m0
m0
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Geostatistical Inversion Geostatistical regularisation (Kitanidis, 1995)
parameters considered spatially correlated random variables deterministic trend Xβ plus random deviation Y′: maximise posterior likelihood of parameters minimise objective function:
Penalise not meeting the measurementsDeviation from prior mean valuePenalise spatial fluctuations
'1
θ|YY'*1
ββ*
'1ZZ
''
YRYββRββ
β,YZZRβ,YZZθ,Z|β,YT
''
)(T
mTmmL
XβYY '
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
depends onelectrode configuration
Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie
Final Remarks Encouraging results:
larger features in ln(K) resolved physical behaviour guaranteed
Not all details resolved in ln(K) field but also in time curves of electrical signals
include higher order moments as data? Starting experiments in 2-D sandboxapplication to real lab data Necessary extensions for field applications:
3-D different boundary conditions
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